Relation between size of parameters and complexity of model with overfitting
Regarding your first question, this is domain/task specific, and not always the case. My guess of why it happens in your situation (You did not specify your domain, so ill assume it's sometimes outside $(-1,1)$) is that higher order polynomials increase much faster than lower order ones so it may fall under the trap where the coefficients have to be large to handle it. (e.g. if $x=2.0$ then $ 2^5*x^2 = x^7 \rightarrow 32*x^2 = x^7$)  

Now to answer why more points will cause less overfitting is sheerly because you have more equations. Your loss is generally some form of average distance accross points and predictions. Given $N$ points I can always find an $N-1$ order polynomial that fits it exactly. Instead of a formal proof I will provide a form in which you can easily generate this polynomial from $N$ $(x_i, y_i)$ pairs
$$ f(x) = \sum_{i=1}^{N}y_i\prod_{j=1\atop j \ != i}^N\frac{(x-x_j)}{(x_i-x_j)}$$  

As you can see, for each $x_i$ it will zero out all terms in the sum except one where the product become 1 making $f(x_i) = y_i$ by construction. So adding more points will prevent whatever optimization process you are learning to directly solve each point exactly, but instead try to find an algorithm that best generalizes to each of the points.